Soap, warm water and a brush. Both are laminates that ignore water, resist most solvents and need no oiling or conditioning at all. The only real difference in care is that Micarta develops a patina from skin oils over years and G10 stays exactly as it was.
G10 is fiberglass cloth in epoxy; Micarta is linen, canvas or paper in phenolic resin. Both are dimensionally stable, waterproof for practical purposes, and immune to rot.
There is nothing to feed, seal or protect. Cleaning is the whole of the maintenance.
Aggressive G10 texture traps grime. An old toothbrush with warm soapy water gets into it, and isopropyl alcohol handles stubborn residue like adhesive or sap.
Rinse and dry afterwards, mostly so water does not sit in the pivot rather than because the handle minds.
Micarta absorbs a little oil from your hands and darkens over years, especially in natural and tan colors. Many owners consider that the point. If you would rather not, a light sanding takes it back.
A slightly worn Micarta handle also grips better when damp than a freshly polished one does.